# Some Notes on the Addition of Interactive Fuzzy Numbers

## Abstract

This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say *A* and *B*, is described by a joint possibility distribution *J*. One can define a fuzzy number \(A +_J B\) (or \(A -_J B\)), called *J*-interactive sum (or difference) of *A* and *B*, in terms of the sup-*J* extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers *B* and *C*, is there a fuzzy number *X* and a joint possibility distribution *J* of *X* and *B* such that \(X +_J B = C\)? (2) Given fuzzy numbers *A*, *B*, and *C*, is there a joint possibility distribution *J* of *A* and *B* such that \(A +_J B = C\)? (3) Given a joint possibility distribution *J* of fuzzy numbers *A* and *B*, is there a joint possibility distribution *N* of \((A +_J B)\) and *B* such that \((A +_J B) -_N B = A\)? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers *A*, *B* and *C* are real numbers, since the fuzzy arithmetic \(+_J\) and \(-_N\) are extension of the classical arithmetic for real numbers.

## Notes

### Acknowledgements

This work was partially supported by FAPESP under grant no. 2016/26040-7 and CNPq under grants no. 306546/2017-5 and 142414/2017-4.

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